rational function - Rational function extra problems 3x3...

Rational function extra problems:
8.
Z
3 x
3
+ 18 x
2
+ 33 x - 2
( x 2 + 6 x + 10) 2
dx
Begin using integration by parts:
3 x 3 + 18 x 2 + 33 x - 2
( x
2
+ 6 x + 10)
2
=
Ax + B
x
2
+ 6 x + 10
+
Cx + D
( x
2
+ 6 x + 10)
2
3 x
3
+ 18 x
2
+ 33 x - 2 = ( Ax + B )( x
2
+ 6 x + 10) + Cx + D
3 x 3 + 18 x 2 + 33 x - 2 = ( A ) x 3 + (6 A + B ) x 2 + (10 A + B + C ) x + (10 B + D )
So we have a system of equations:
A=3
6A+B=18
10A+6B+C=33
10B+D=-2
So we have that A=3, B=0, C=3, D=-2
Thus
Z
3 x 3 + 18 x 2 + 33 x - 2
( x 2 + 6 x + 10) 2
dx =
Z
3 x
x 2 + 6 x + 10
+
3 x - 2
( x 2 + 6 x + 10) 2
dx
Next we can complete the square to see that x
2
+6 x +10 = ( x +3)
2
+1 and use u substitution
where u=x+3,x=u-3, du=dx
So we get 3
Z
u - 3
u
2
+ 1
du +
Z
3 u - 11
( u
2
+ 1)
2
du = 3
Z
u
u
2
+ 1
du - 3
Z
1
u
2
+ 1
du + 3
Z
u
( u
2
+ 1)
2
du -
11
Z
1
( u
2
+ 1)
2
du we can solve several of these with simple substitution v = u 2 + 1 , dv = 2 udu
1

some we need a trig substitution of u = tanθ, du = sec 2 θdθ
3
Z
u
u
2
+ 1
du - 3
Z
1
u
2
+ 1
du + 3
Z
u
( u
2
+ 1)
2
du - 11
Z
1
( u
2
+ 1)
2
du
=
3
2
Z
1
v
dv - 3
Z
1
u
2
+ 1
du +
3
2
Z
1
v
2
dv - 11
Z
sec 2 θ
(tan 2 θ + 1)
2
dθ
=
3
2
ln | v | - 3 arctan( u ) -
3
2
1
v
- 11
Z
sec
2
θ
sec 4 θ
dθ
=
3
2
ln | u
2
+ 1 | - 3 arctan( u ) -
3
2
1
( u
2